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Chapter 2: Problem 25

Perform the operation and write the result in standard form.$$(9-i)-(8-i)$$

Short Answer

Expert verified
The result of the operation is 1.

Step by step solution

01

Identify the Real and Imaginary Parts

The first complex number is \(9-i\), so the real part is 9 and the imaginary part is -1. The second complex number is \(8-i\), so the real part is 8 and the imaginary part is -1.
02

Subtract the Real Parts and the Imaginary Parts

Subtract the real parts: \(9 - 8 = 1\) and subtract the imaginary parts: \(-1 - (-1) = 0\).
03

Write the Result in Standard Form

The result is \(1 + 0i\), but since the imaginary part is 0, we can simply write the result as 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Real Part
In the realm of complex numbers, the real part is a crucial component. A complex number is typically expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers.
\(a\) denotes the real part of the complex number.
  • The real part is the portion without the imaginary unit \(i\).
  • In our example with the complex numbers \(9-i\) and \(8-i\), the real parts are 9 and 8, respectively.
  • When performing operations such as addition or subtraction with complex numbers, manage the real parts separately from the imaginary parts.
In the given exercise, subtract the real parts: 9 (from \(9-i\)) minus 8 (from \(8-i\)), which equals 1.
Imaginary Part
The imaginary part of a complex number is represented by the coefficient of the imaginary unit \(i\). While it might sound complex, it's actually quite straightforward!
  • For a complex number \(a + bi\), the imaginary part is the \(b\) term.
  • In the exercise, both complex numbers, \(9-i\) and \(8-i\), have an imaginary part of \(-1\).
  • Just like with real parts, when performing arithmetic operations, separate the imaginary parts as their own entities.
In this example, subtract the imaginary parts, resulting in \(-1 - (-1) = 0\). This is essential for correctly manipulating complex numbers.
Standard Form
Writing a complex number in standard form means expressing it as \(a + bi\) where \(a\) is the real part and \(bi\) is the imaginary part. The aim is to keep both parts distinctly separate and visible.
A complex number with no imaginary component is simply a real number.
  • If the imaginary part is zero, as in the final result here, you can omit it.
  • In our exercise, after subtracting \(9-i\) by \(8-i\), we simplify to \(1 + 0i\).
  • Thus, the standard form of the result is simply 1.
Knowing how to express complex numbers in standard form is crucial for clarity and ease when performing mathematical operations.

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